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Международная молодежная конференция «Геометрия и управление»
18 апреля 2014 г. 10:50, г. Москва, МИАН

Metric Geometry of Carnot–Carathéodory Spaces and Its Applications

Maria Karmanova

Sobolev Institute of Mathematics, Novosibirsk, Russia
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Maria Karmanova

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Аннотация: We describe new fine properties of Carnot–Carathéodory spaces under minimal assumptions on smoothness of the basis vector fields. As a corollary, we discover new geometric properties of weighted Carnot-Caratheodory spaces. All these results are new even for a “smooth” case. They play crucial role in the development of the differentiability theory on sub-Riemannian structures (see, e.g., works by S. Vodopyanov [1, 2]), in the investigation of non-equiregular Carnot–Carathéodory spaces (see, e.g., work by S. Selivanova [3]) and imply many basic results of the theory of non-holonomic spaces (see, e.g., work by S. Basalaev and S. Vodopyanov [4]).
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Definition (see, e.g., [5, 2, 4, 6, 7]). Fix a connected Riemannian $C^{\infty}$-manifold $\mathbb M$ of topological dimension $N$. The manifold $\mathbb M$ is called the Carnot–Carathéodory space if the tangent bundle $T\mathbb M$ has a filtration
$$ H\mathbb M=H_1\mathbb M\subsetneq\ldots\subsetneq H_i\mathbb M\subsetneq\ldots\subsetneq H_M\mathbb M=T\mathbb M $$
by subbundles such that every point $p\in\mathbb M$ has a neighborhood $U\subset\mathbb M$ equipped with a collection of $C^{1}$-smooth vector fields $X_1,…,X_N$ enjoying the following two properties.
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(1) At every point $v\in U$ we have a subspace
$$ H_i\mathbb M(v)=H_i(v)=\operatorname{span}\{X_1(v),…,X_{\dim H_i}(v)\}\subset T_v\mathbb M $$
of the dimension $\dim H_i$ independent of $v$, $i=1,\ldots,M$.
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(2) The inclusion $[H_i, H_j]\subset H_{i+j}$, $i+j\leq M$, holds.
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Moreover, if the third condition holds then the Carnot–Carathéodory space is called the Carnot manifold:
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(3) $H_{j+1}=\operatorname{span}\{H_j, [H_1,H_{j}], [H_2,H_{j-1}],\ldots,[H_k, H_{j+1-k}]\}$, where $k=[\frac{j+1}{2}]$, $H_0=\{0\}$, $j=1,\ldots, M-1$.
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The subbundle $H\mathbb M$ is called horizontal.
The number $M$ is called the depth of the manifold $\mathbb M$.
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The main result is the following
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Theorem [6, 7]. Let $\mathbb M$ be a Carnot–Carathéodory space with $C^{1,\alpha}$-smooth basis vector fields, $\alpha\geq0$ (if $\alpha=0$ then the fields belong to the class $C^1$). Then for each point of $\mathbb M$, there exists a sufficiently small neighborhood $\mathcal U\Subset\mathbb M$ possessing the following property: for $u,v\in \mathcal U$, $w=\gamma(1)$ and $\widehat{w}=\widehat{\gamma}(1)$, where $\gamma,\widehat{\gamma}:[0,1]\to\mathbb M$ are absolutely continuous (in the classical sense) curves contained in $\operatorname{Box}(u, \varepsilon)$ such that
\begin{equation*} \dot{\gamma}(t)=\sum\limits_{i=1}^Nb_i(t)X_i(\gamma(t)), \gamma(0)=v, \quadand\quad \dot{\widehat{\gamma}}(t)=\sum\limits_{i=1}^Nb_i(t)\widehat{X}^u_i(\gamma(t)), \widehat{\gamma}(0)=v, \end{equation*}
and each measurable function $b_i(t)$ meets the property
$$ \tag{1} \int_0^1|b_i(t)| dt<S\varepsilon^{\operatorname{deg}X_i}, $$
$S<\infty$, $i=1,\ldots, N$, we have
$$ \tag{2} \max\{d_{\infty}(w,\widehat{w}), d^u_{\infty}(w,\widehat{w})\}= \begin{cases} O(1)\cdot\varepsilon^{1+\frac{\alpha}{M}} if \alpha>0,
o(1)\cdot\varepsilon if \alpha=0, \end{cases} $$
with $O(1)$ and $o(1)$ to be uniform in $u\in\mathcal U$ and all collections of functions $\{b_i(t)\}_{i=1}^N$ with the property (1).
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Remark [7] (see also [8]). For weighted Carnot–Carathéodory spaces, the estimate in (2) is $O(1)\cdot\varepsilon^{1+\frac{\alpha}{l_M}}$ for $\alpha>0$, where $l_M$ is the maximal weight [3, 7, 8].
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The research is supported by Grant of the Government of Russian Federation for the State Support of Researches (Agreement No 14.B25.31.0029).

Материалы: slides.pdf (952.8 Kb), abstract.pdf (94.7 Kb)

Язык доклада: английский

Список литературы
  1. Vodopyanov S. K., “Geometry of Carnot–Carathéodory spaces and differentiability of mappings”, The Interaction of Analysis and Geometry, Amer. Math. Soc. Providence, 2007, 247–302  mathscinet
  2. Karmanova M., Vodopyanov S. Geometry of Carnot–Carathéodory spaces, differentiability, coarea and area formulas, Analysis and Mathematical Physics. Trends in Mathematics, Birkhauser, Basel, 2009, 233–335  mathscinet  zmath
  3. Selivanova S., “Metric geometry of nonregular weighted Carnot–Carathéodory spaces”, Journal of Dynamical Control Systems, 20 (2014), 123–148  crossref  mathscinet  zmath  isi  scopus
  4. Basalaev S.G., Vodopyanov S.K., “Approximate differentiability of mappings of Carnot–Carathéodory spaces”, Eurasian Math. J., 4:2 (2013), 10–48  mathnet  mathscinet  zmath
  5. Gromov M., “Carnot–Carathéodory spaces seen from within”, Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, 79–323  mathscinet  zmath
  6. Karmanova M., Vodopyanov S., “On local approximation theorem on equiregular Carnot-Caratheodory spaces”, Proc. INDAM Meeting on Geometric Control and Sub-Riemannian Geometry, Cortona, May 2012, 5, Springer, 2014, 241–262, INDAM Ser.  crossref  mathscinet  zmath
  7. Karmanova M., “Fine properties of basis vector fields of Carnot–Carathéodory spaces under minimal assumptions on smoothness”, Siberian Mathematical Journal, 55:1 (2014), 87–99  mathnet  crossref  mathscinet  zmath  isi  scopus
  8. Karmanova M., “Fine properties of weighted Carnot–Carathéodory spaces under minimal assumptions on smoothness”, Ann. Univ. Bucharest (Math. Ser.), 2014 \toapprear  mathscinet

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